Testing parametric models for the angular measure for bivariate extremes

TL;DR

This paper develops a goodness-of-fit test for parametric models of the angular measure in bivariate extreme value analysis, using a Wasserstein distance-based statistic and bootstrap methods, with applications to river discharge data.

Contribution

It introduces a novel test statistic based on the Wasserstein distance between nonparametric and parametric estimators of the angular measure, including asymptotic theory and bootstrap validation.

Findings

The test performs well in finite samples for logistic and Hüsler--Reiss models.

The method effectively detects model misspecification in simulated data.

Application to river data demonstrates practical utility.

Abstract

The angular measure on the unit sphere characterizes the first-order dependence structure of the components of a random vector in extreme regions and is defined in terms of standardized margins. Its statistical recovery is an important step in learning problems involving observations far away from the center. In this paper, we test the goodness-of-fit of a given parametric model to the extremal dependence structure of a bivariate random sample. The proposed test statistic consists of a weighted $L_1$-Wasserstein distance between a nonparametric, rank-based estimator of the true angular measure obtained by maximizing a Euclidean likelihood on the one hand, and a parametric estimator of the angular measure on the other hand. The asymptotic distribution of the test statistic under the null hypothesis is derived and is used to obtain critical values for the proposed testing procedure via a parametric bootstrap. Consistency of the bootstrap algorithm is proved. A simulation study illustrates the finite-sample performance of the test for the logistic and H\"usler--Reiss models. We apply the method to test for the H\"usler--Reiss model in the context of river discharge data.